288 Frequently Asked Questions In Quantitative Finance
The floorlet can be thought of in a similar way in terms
of a put on the forward rate and so its formula is
e−r(Ti+^1 −t)(KN(−d 2 )−FN(−d 1 )).
Swaptions A payer swaption, which is the right to pay
fixed and receive floating, can be modelled as a call on
the forward rate of the underlying swap. Its formula
is then
1 −(^1
1 +mF
)τm
F
e−r(T−t)(FN(d 1 )−KN(d 2 )),
whereris the continuously compounded interest rate
applicable fromttoT, the expiration,Fis the forward
swap rate,Kthe strike and
d 1 =
ln(F/K)+^12 σ^2 (T−t)
σ
√
T−t
,
d 2 =
ln(F/K)−^12 σ^2 (T−t)
σ
√
T−t
,
whereσis the volatility of the forward swap rate.τis
the tenor of the swap andmthe number of payments
per year in the swap.
The receiver swaption is then
1 −(^1
1 +mF
)τm
F
e−r(T−t)(KN(−d 2 )−FN(−d 1 )).
Spot rate models
The above method for pricing derivatives is not entirely
internally consistent. For that reason there have been
developed other interest rate models that are internally
consistent.
